Separation of Variables

Separation of variables is the main method by which three-dimensional Schrödinger problems become solvable. One looks for product solutions whose coordinate dependence factors into simpler pieces, reducing a partial differential equation to ordinary differential equations tied together by separation constants.

Separation is not automatic. It works only when the Hamiltonian, coordinate system, and boundary conditions have compatible structure.

Stationary Separation

For a time-independent Hamiltonian, begin with

$$\left[ -\frac{\hbar^2}{2m}\nabla^2 +V(\mathbf r) \right]\varphi(\mathbf r) =E\varphi(\mathbf r).$$

The first separation is often time from space:

$$\psi(\mathbf r,t) =\varphi(\mathbf r)e^{-iEt/\hbar}.$$

The remaining question is whether the spatial function $\varphi(\mathbf r)$ can be factored further.

In Cartesian coordinates one tries

$$\varphi(x,y,z)=X(x)Y(y)Z(z).$$

In spherical coordinates one tries

$$\varphi(r,\theta,\phi)=R(r)\Theta(\theta)\Phi(\phi),$$

or, more compactly for central potentials,

$$\varphi(r,\theta,\phi)=R_{n\ell}(r)Y_\ell^m(\theta,\phi).$$

The notation changes with the problem, but the logic is the same: product structure turns one multi-variable equation into several one-variable equations.

Cartesian Example

Suppose

$$V(x,y,z)=V_x(x)+V_y(y)+V_z(z).$$

In Cartesian coordinates,

$$\nabla^2 =\partial_x^2+\partial_y^2+\partial_z^2.$$

Try $\varphi=XYZ$. Dividing by $XYZ$ gives

$$-\frac{\hbar^2}{2m}\frac{X''}{X} +V_x(x) -\frac{\hbar^2}{2m}\frac{Y''}{Y} +V_y(y) -\frac{\hbar^2}{2m}\frac{Z''}{Z} +V_z(z) =E.$$

Each coordinate-dependent part must be constant. Write

$$\left[ -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V_x(x) \right]X=E_xX,$$

and similarly for $Y$ and $Z$. The total energy is

$$E=E_x+E_y+E_z.$$

This is why rectangular boxes and separable oscillator potentials reduce to one-dimensional problems.

Spherical Separation

For a central potential,

$$V(\mathbf r)=V(r),$$

spherical coordinates match the symmetry. The angular part separates into spherical harmonics:

$$Y_\ell^m(\theta,\phi), \qquad \ell=0,1,2,\ldots, \qquad m=-\ell,\ldots,\ell.$$

The separation constants are angular-momentum quantum numbers. The remaining radial equation contains an effective centrifugal term depending on $\ell$. The compact central-potential derivation is Angular and Radial Separation.

This is the route to central potentials and the hydrogen atom. The full radial equation has its own canonical page later in this volume.

Cylindrical Preview

For potentials with cylindrical symmetry, coordinates $(\rho,\phi,z)$ may separate:

$$\varphi(\rho,\phi,z) =R(\rho)\Phi(\phi)Z(z).$$

Single-valuedness around the azimuthal angle gives

$$\Phi(\phi)=e^{im\phi}, \qquad m\in\mathbb Z.$$

Cylindrical separation appears in waveguides, magnetic-field problems, and systems with an axis of symmetry. The method is the same, but the radial equations often involve Bessel functions rather than sines, cosines, or spherical harmonics.

Separation Constants As Quantum Numbers

Separation constants become quantum labels only after boundary conditions and normalizability are imposed. In a box, they become integer mode numbers. In a central potential, they become angular-momentum labels. In scattering, they can label partial waves or continuous momenta.

The workflow is:

1. choose coordinates adapted to the potential and boundaries;
2. try a product ansatz;
3. divide by the product and isolate coordinate-dependent terms;
4. introduce separation constants;
5. solve the resulting ordinary differential equations;
6. impose boundary conditions and normalization;
7. superpose separated modes to build general states.

The constants are not arbitrary decorations. They encode compatibility between the differential equation and the allowed boundary behavior.

Product Modes And Superposition

A separated solution is usually one mode, not the most general state. For a discrete spectrum, a general state may be expanded as

$$\psi(\mathbf r,t) =\sum_\lambda c_\lambda \varphi_\lambda(\mathbf r) e^{-iE_\lambda t/\hbar},$$

where $\lambda$ denotes the full set of quantum labels. For a three-dimensional box, $\lambda=(n_x,n_y,n_z)$. For a central potential, $\lambda$ may include $(n,\ell,m)$.

For continuous spectra, sums are replaced or supplemented by integrals. Completeness and convergence are part of the mathematical content of the separated basis.

When Separation Fails

Separation can fail because:

• the potential does not split in the chosen coordinates;
• the boundary conditions mix coordinates;
• the region has a geometry not aligned with the coordinate surfaces;
• the Hamiltonian includes terms coupling coordinates or internal states;
• the desired state is easier to describe by numerical methods than by analytic modes.

Failure to separate does not mean the problem is physically invalid. It means a different method is needed.

Common Mistakes

• Trying Cartesian separation for a spherically symmetric problem and missing the simpler angular structure.
• Assuming product solutions alone are the full solution, rather than a basis to be superposed.
• Treating separation constants as quantum numbers before imposing boundary conditions.
• Forgetting that boundary conditions must separate too.
• Dropping coordinate factors from the Laplacian in spherical or cylindrical coordinates.
• Using the same symbol for different separation constants without tracking their physical meaning.

Where This Is Used

• Schrödinger Equation in Three Dimensions sets up the 3D equation.
• Three-Dimensional Box is the first Cartesian product example.
• Free Particle in Three Dimensions gives the continuum plane-wave basis and energy-shell picture.
• Spherical Coordinates gives the coordinate formulas used for central potentials.
• Angular and Radial Separation carries out the central-potential split into angular and radial equations.
• Degeneracy in Separable Systems explains repeated energies that arise after separated quantum numbers are introduced.
• Propagators in Multiple Dimensions gives the time-domain product kernel associated with separable Hamiltonians.
• Time-Independent Schrödinger Equation explains the stationary eigenvalue problem.
• Spherical Harmonics supplies the angular functions used in central potentials.
• Separation of Variables gives the mathematical method in broader PDE language.

References

• D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed., Cambridge University Press, 2018.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, Wiley, 1977.
• G. B. Arfken, H. J. Weber, and F. E. Harris, Mathematical Methods for Physicists, 7th ed., Academic Press, 2012.

Exercises

1. Suppose $V(x,y,z)=V_x(x)+V_y(y)+V_z(z)$. Show that product solutions have additive energies.

Solution

With $\varphi=XYZ$, substitution gives

$$\left[ -\frac{\hbar^2}{2m}\frac{X''}{X} +V_x(x) \right] + \left[ -\frac{\hbar^2}{2m}\frac{Y''}{Y} +V_y(y) \right] + \left[ -\frac{\hbar^2}{2m}\frac{Z''}{Z} +V_z(z) \right] =E.$$

Each bracket depends on only one coordinate, so each equals a constant: $E_x$, $E_y$, and $E_z$. Therefore

$$E=E_x+E_y+E_z.$$

2. Why are spherical coordinates natural for $V(\mathbf r)=V(r)$?

Solution

The potential depends only on distance from the origin and is independent of angle. Spherical coordinates separate the radial coordinate from angular coordinates, so the angular dependence can be organized by spherical harmonics and the remaining dynamics reduced to a radial equation.

3. Give one reason separation may fail even if the potential looks simple.

Solution

Boundary conditions can prevent separation. For example, a potential may be simple in Cartesian coordinates, but if the allowed region has a tilted or curved boundary that mixes $x$, $y$, and $z$, a product ansatz $X(x)Y(y)Z(z)$ may not satisfy the boundary conditions.